Is it possible to find out the time of the viewer's connection to the chat?

Not accurately no

Join/part events are batched, and don’t work in large channels.

So you are limited to the undocumented TMI/chatters API at https://tmi.twitch.tv/group/user/CHANNELNAME/chatters it’s undocumented and can change/break at any time, so use at your own risk.

The other alternative is to watch chat for messages, but that excludes lurkers.

Alternatively a video extension may have here, with a socket connection or similar.

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